ondřej souček supervizor: prof. rndr. zdeněk martinec, drsc
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Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Numerical modeling of ice-sheet dynamics
Ondřej Souček
supervizor: Prof. RNDr. Zdeněk Martinec, DrSc.
Department of Geophysics, Faculty of Mathematics and Physics, Charles
University in Prague
Research Institute of Geodesy, Topography and Cartography, Zdiby
1.6.2010
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Overview
Ice-sheets and their dynamics
Continuum thermo-mechanical model of a glacier
The Shallow Ice approximation
The SIA-I iterative algorithm
Numerical benchmarks
ISMIP-HOM A exp. - steady-stateISMIP-HOM F exp. - prognosticEISMINT - Greenland Ice Sheet models
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scheme of an ice sheet
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Transport processes in an ice sheet
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Simplified ice sheet model
Cold ice
Lithosphere
Temp. ice
z
x,y
Atmospheres
CTS
b
F (x,t) = 0
F (x,t) = 0F (x,t) = 0
Cold-ice zone (pure ice only)
Temperate-ice zone (liquid water present)
Free surface and glacier base represented by differentiable surfaces
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Modelled problem - cold ice zone
Stokes’ flow problem
divτ + ρ~g = ~0
Equation of continuity (Constanthomogeneous ice density)
div~v = 0
Energy balance – Heat transportequation
ρc(T )T = div(κ(T )gradT )+τ.. ε
Rheology
τij = −pδij + σij
σij = 2ηεij
εij =1
2
(
∂vi
∂xj
+∂vj
∂xi
)
η =1
2A(T ′)−
13 ε
− 23
II
A(T ′) = A0 exp
(
−
Q
R(T0 + T ′)
)
εII =
√
˙εkl ˙εkl
2
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Boundary conditions
Free surface
Kinematic boundary condition
∂fs
∂t+ ~v · gradfs = as
Dynamic boundary condition
Traction-free
τ · ~ns = ~0
Surface temperature
T = Ts(~x, t)
Accumulation-ablation function
as= a
s(~x, t)
Glacier base
Kinematic boundary condition fb (~x, t) given
Dynamic boundary conditions
Frozen-bed conditions, T < TM :
~v = ~0
Sliding law T = TM :
~v = ~g(τ · ~n)
Geothermal heat flux
~q+= ~q
geo(~x, t)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scaling Approximation – plausibility and motivation
Glaciers and ice sheets are typically very flat features, with the aspect(height:horizontal) ratio less than 1
10for small valley glaciers and one
order less for big ice sheets ( 1100
)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scaling Approximation – plausibility and motivation
Natural scaling may be introduced:kinematic quantities
(x1, x2, x3) = ([L]x1, [L]x2, [H]x3)
(v1, v2, v3) = ([V↔]v1, [V↔]v2, [Vl]v3)
(fs(x1, x2), fb(x1, x2)) = [H](fs(x1, x2), fb(x1, x2))
ε =[H]
[L]=
[Vl]
[V↔]
dynamic quantities -
p = ρg [H]p
(σ13, σ23) = ρg [H](σ13, σ23)
(σ11, σ22, σ12) = ρg [H](σ11, σ22, σ12)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scaling Approximation – plausibility and motivation
Natural scaling may be introduced:kinematic quantities
(x1, x2, x3) = ([L]x1, [L]x2, [H]x3)
(v1, v2, v3) = ([V↔]v1, [V↔]v2, [Vl]v3)
(fs(x1, x2), fb(x1, x2)) = [H](fs(x1, x2), fb(x1, x2))
ε =[H]
[L]=
[Vl]
[V↔]
dynamic quantities - Shallow Ice Approximation SIA
p = ρg [H]p
(σ13, σ23) = ερg [H](σ13, σ23)
(σ11, σ22, σ12) = ε2ρg [H](σ11, σ22, σ12)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scaling Approximation – plausibility and motivation
Standard scaling perturbation series constructed: all field unknowns(vi , σij , fs , fb) are expressed by a power series in the aspect ratio ε
q = q(0) + εq
(1) + ε2q(2) + . . .
and inserting these expressions into governing equations (eq. of motion,continuity, rheology) leads to separation of these equations according to theorder of ε. (Implicit assumption: not only q is now scaled to unity but also itsgradient (???))
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scaling Approximation – plausibility and motivation
Keeping only lowest-order terms, we arrive at the Shallow Ice Approximation,which can be explicitly solved
p(0)(·, x3) = f
(0)s (·)− x3
σ(0)13 (·, x3) = −
∂f(0)s (·)
∂x1(f (0)
s (·)− x3)
σ(0)23 (·, x3) = −
∂f(0)s (·)
∂x2(f (0)
s (·)− x3)
where (·) ≡ (x1, x2)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scaling Approximation – plausibility and motivation
Also the velocities may be expressed semi-analytically as
v(0)1 (·, x3) = v
(0)1 (·)sl +
∫ x3
f(0)
b(·)
A(T ′(·, x ′3))(
σ(0)13
2+ σ
(0)23
2)
σ(0)13 (·, x
′3)dx
′3
v(0)2 (·, x3) = v
(0)2 (·)sl +
∫ x3
f(0)
b(·)
A(T ′(·, x ′3))(
σ(0)13
2+ σ
(0)23
2)
σ(0)23 (·, x
′3)dx
′3
v3 from equation of continuity
v(0)3 (·, x3) = v
(0)3 (·)sl −
∫ x3
f(0)
b(·)
(
∂v(0)1
∂x1(·, x ′
3) +∂v
(0)2
∂x2(·, x ′
3)
)
dx′3
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Problems of SIA
Zeroth order model – looses validity in regions where higher-order terms
become important:
Regions of high curvature of the surface
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Problems of SIA
Zeroth order model – looses validity in regions where higher-order terms
become important:
Regions of high curvature of the surfaceRegions where a-priori dynamic scaling assumptions areviolated (floating ice – SSA, ice streams)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Problems of SIAZeroth order model – looses validity in regions where higher-order terms
become important:
Regions of high curvature of the surfaceRegions where a-priori dynamic scaling assumptions areviolated (floating ice – SSA, ice streams)
Figure: RADARSAT Antarctic Mapping Project
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Solution?
Higher-order models – continuation of the expansion procedure andsolving for higher-order corrections (Pattyn F., Rybak O.)
Full Stokes Solvers – Finite Elements Methods (Elmer - Gagliardini O.),Spectral methods (Hindmarsch R.)
PROBLEM is the speed of full-Stokes and higher-order techniques, thecomputational demands disable usage of these techniques in large-scaleevolutionary models
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Any ideas?
Aim:
Find an "intermediate" technique that would exploit the scalingassumptions of flatness of ice but would provide "better" solution thanSIA:
KEY IDEA: Apply the scaling assumptions of smallness not on theparticular stress components but only on their deviations from thefull-Stokes solution
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
SIA-I
~un //δ~un+ 1
2// ~un+ 1
2 = ~un + ǫ1δun+ 1
2
��
~un+1 = (1 − ǫ2)~un+ 1
2 + ǫ2~u∗n+ 1
2 ~u∗n+ 12
oo~vn+ 1
2oo
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Ice-Sheet Model Intercomparison Project - Higher Order
Models (ISMIP-HOM) - experiment A
fs (x1, x2) = −x1 tanα , α = 0.5◦,
fb(x1, x2) = fs (x1, x2) − 1000
+ 500 sin(ωx1) sin(ωx2)
with
ω =2π
[L].
Ice considered isothermal
Boundary conditions
Free surfaceFrozen bedAt the sides, periodic boundaryconditions are prescribed:
~v(x1, 0, x3) = ~v(x1, [L], x3)
~v(0, x2, x3) = ~v([L], x2, x3)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
ISMIP - HOM experiment A
0
20
40
60
80
100
120
0 0.2 0.4 0.6 0.8 1
v x (
m a
-1)
at f
s
x
30
40
50
60
70
80
90
100
110
120
0 0.2 0.4 0.6 0.8 1
σ xz
[kPa
] at
fb
x
Figure: Pattyn F., ISMIP-HOM results preliminary reportOndřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
ISMIP - HOM experiment A
0
20
40
60
80
100
120
0 0.2 0.4 0.6 0.8 1
v x (
m a
-1)
at f
s
x
30
40
50
60
70
80
90
100
110
120
0 0.2 0.4 0.6 0.8 1
σ xz
[kPa
] at
fb
x
Figure: Pattyn F., ISMIP-HOM results preliminary reportOndřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
ISMIP - HOM experiment A
Computational detailsResolution: 41 × 41 × 41
Number of iterations: ε = 120 ∼ 40 iter., ε = 1
10 ∼ 100 iter.
Each iter. step took approximately 0.22 s at Pentium 4, 3.2.GHz
For comparison Elmer (Gagliardini) Full-Stokes solver ∼ 104 CPU s
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
ISMIP-HOM experiment F
Experiment description - ice slab flowingdownslope (3◦) over a Gaussian bump
Linear rheology!
Output: Steady state free surface profileand velocities
Comparison with ISMIP-HOMfull-Stokes finite-element solution(Olivier Gagliardini computing by Elmer)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
ISMIP-HOM experiment F
Free surface (left - Elmer, right- SIA-I)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
ISMIP-HOM experiment F
Surface velocities
Numerical performance
Figure: Gagliardini & Zwinger, 2008
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
ISMIP-HOM experiment FSurface velocities (left-Elmer,right-SIA-I)
Numerical performance
0
2
4
6
8
10
12
0 20 40 60 80 100 120 140 160 180 200C
PU-t
ime/
time-
step
[s]
Iteration
’time_demands.dat’
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Extension for non-linear rheology
We take setting from ISMIP-HOMexperiment A (L=80) - flow of aninclined ice slab over a sinusoidal bump,periodically elongated
Evolution of free surface, steady state
Impossible to compare with evolution inElmer (too CPU time demanding)
Time demands of the SIA-I algorithmpractically the same as for the linearcase !!!
Let’s compare only velocities atparticular time instants
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Extension for non-linear rheology
t = 50a (left - Elmer, right - SIA-I) t = 100a (left - Elmer, right - SIA-I)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT benchmark - Greenland Ice Sheet Models
Paleoclimatic simulation
Prognostic experiment - global warming scenario
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
Two glacial cycles (cca 250 ka)
Temperature + sea-level forcing based on ice-core δ18O isotope record
-14-12-10-8-6-4-2 0 2 4 6
-250 -200 -150 -100 -50 0
∆ T
(K
)
time (ka)
-140
-120
-100
-80
-60
-40
-20
0
20
-250 -200 -150 -100 -50 0
∆ Se
a le
vel (
m)
time (ka)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
0
0
0
0
1
1
1
1
2
2
2
3
3
3
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Surface topography (km), age = 150 ka
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
0
0
0
0
1
1
1
1
2
2
2
3
3
3
0.0
0.5
1.0
1.5
2.0
2.5
y [1
03 km
]
0.0 0.5 1.0 1.5
x [103 km]
Surface topography [km], age = 135 ka
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
0
0
0
0
0
1
1
1
1
2
2
2
2
3
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Surface topography (km), age = 125 ka
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
0
0
0
0
0
1
11
1
2
2
2
2
3
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Surface topography (km), age = 115 ka
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
0
0
00
1 1
1
1
1
2
2
2
33
3
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Surface topography (km), age = 75 ka
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
0
0
0
0
1
1
1
1
2
2
2
3
3
3
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Surface topography (km), age = 0 a
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
-250 -200 -150 -100 -50 0
Gla
ciat
ed a
rea
(106 k
m2 )
time (ka)
1.5
2
2.5
3
3.5
4
4.5
5
-250 -200 -150 -100 -50 0
Ice-
shee
t vol
ume
(106 k
m3 )
time (ka)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
(Huybrechts, 1998)
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
-250 -200 -150 -100 -50 0
Gla
ciat
ed a
rea
(106 k
m2 )
time (ka)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Prognostic experiment
Model response to an artificial warming scenario
temperature increase by 0.035◦ C per year for the first 80 years (total2.8◦ C increase)
by 0.0017◦ per year C for the remaining 420 years (0.714◦ C)
In total temperature increase of 3.514◦ C within 500 years
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Prognostic experiment
0
0
0
0
1
1
1
1
2
2
2
3
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Surface topography (km), t = 0 a
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Prognostic experiment
0
0
0
0
1
1
1
1
2
2
2
3
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Surface topography (km), t = 0 a
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−0.4
−0.4
−0.4
0.4
0.4
0.4
0.8
0.8
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Initial accumulation−ablation function (m a −1)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Prognostic experiment
0
0
0
0
1
1
1
1
2
2
2
3
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Surface topography (km), t = 0 a
0
0
0
0
1
1
1
1
2
2
2
3
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Surface topography (km), t = 500 a
−12−11.6
−11.2−10.8
−10.4
−10
−10
−10
−9.6
−9.6
−9.2
−9.2
−8.8
−8.8
−8.4
−8.4
−8.4
−8
−8 −8
−7.6
−7.6 −7.6
−7.2
−7.2
−7.2
−6.8
−6.8
−6.8
−6.4
−6.4
−6.4
−6
−6
−6
−6
−6
−5.6
−5.6
−5.6
−5.6
−5.6
−5.2
−5.2
−5.2
−5.2
−5.2 −5.2
−4.8
−4.8
−4.8
−4.8
−4.8 −4.8
−4.4
−4.4
−4.4
−4.4
−4.4
−4.4
−4
−4
−4
−4
−4
−4
−4
−3.6−3.6
−3.6
−3.6
−3.6−3.6
−3.6 −3.6
−3.2
−3.2
−3.2
−3.2
−3.2
−3.2
−3.2
−3.2 −3.2
−2.8
−2.8
−2.8
−2.8
−2.8
−2.8
−2.8−2.8
−2.8
−2.8
−2.4−2.4
−2.4
−2.4
−2.4
−2.4
−2.4
−2.4
−2.4
−2.4
−2
−2−2
−2−2
−2
−2
−2
−2
−2
−1.6
−1.6−1.6−1.6
−1.6
−1.6
−1.6−1.6
−1.6
−1.6
−1.6
−1.2
−1.2−1.2
−1.2
−1.2
−1.2
−1.2−1.2
−1.2
−1.2
−1.2
−0.8
−0.8
−0.8
−0.8
−0.8
−0.8
−0.8
−0.8−0.8
−0.8
−0.8
−0.4
−0.4
−0.4
−0.4
−0.4
−0.4
−0.4
−0.4
−0.4
−0.4
−0.4
0.4
0.4
0.4
0.8
0.8
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Initial accumulation−ablation function (m a −1)
Ondřej Souček Ph.D. defense
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Prognostic experiment
0
0
0
0
1
1
1
1
2
2
2
3
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Surface topography (km), t = 0 a
0
0
0
0
1
1
1
1
2
2
2
3
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Surface topography (km), t = 500 a
−600
−600 −4
00
−200
−200
−200
−200
−200
0.0
0.5
1.0
1.5
2.0
2.5
y (1
03 km
)
0.0 0.5 1.0 1.5
x (103 km)
Topography difference (m)
−1200−1100−1000
−900−800−700−600−500−400−300−200−100
0100200300400500600700800
Ondřej Souček Ph.D. defense
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